#8139482

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,23,27)(19,28,29,24)(20,21,25,30)(22,31,32,26)(34,38)(35,41)(36,43)(46,48);(3,10)(6,7)(11,16)(14,15)(17,19)(18,24)(20,22)(21,26)(23,29)(25,32)(27,28)(30,31)(33,34,37,38)(35,39,41,45)(36,40,43,44)(42,46,47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,20,23,25)(18,21,27,30)(19,22,29,32)(24,26,28,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,47)(44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,24)(20,21)(22,26)(23,27)(25,30)(28,29)(31,32)(33,34)(35,39)(36,44)(37,38)(40,43)(41,45)(42,48)(46,47);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,26)(20,27)(22,28)(23,30)(24,32)(29,31)(33,36)(34,40)(35,42)(37,43)(38,44)(39,46)(41,47)(45,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,23,32)(18,26,27,31)(19,20,29,25)(21,28,30,24)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046678);